Looks like a compound of a ex with gohi. Cells are connected by further 120 diametral id. Its vertex figure is a variation of qrid, in fact x3o5/3f or x3/2o5/2f (which degenerates to something what looks like an ike plus an inscribed f-sissid, but diametral rectangles will there be in addition). Accordingly edges come coincident by 5, all faces come in coincident pairs, and the id also come in coincident pairs.

As abstract polytope ex+gohi+120id is isomorphic to gax+gashi+120gid, thereby replacing pentagons by pentagrams resp. replacing gad by sissid and id by gid.

Incidence matrix according to Dynkin symbol

x3o3o5/2o5/4*a

. . .   .      | 120 |   60 |   60   60 |  20  30  12
---------------+-----+------+-----------+------------
x . .   .      |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x3o .   .      |   3 |    3 | 2400    * |   1   1   0
x . .   o5/4*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x3o3o   .      ♦   4 |    6 |    4    0 | 600   *   *
x3o .   o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x . o5/2o5/4*a ♦  12 |   30 |    0   12 |   *   * 120

x3o3o5/3o5*a

. . .   .    | 120 |   60 |   60   60 |  20  30  12
-------------+-----+------+-----------+------------
x . .   .    |   2 | 3600 |    2    2 |   1   2   1
-------------+-----+------+-----------+------------
x3o .   .    |   3 |    3 | 2400    * |   1   1   0
x . .   o5*a |   5 |    5 |    * 1440 |   0   1   1
-------------+-----+------+-----------+------------
x3o3o   .    ♦   4 |    6 |    4    0 | 600   *   *
x3o .   o5*a ♦  30 |   60 |   20   12 |   * 120   *
x . o5/3o5*a ♦  12 |   30 |    0   12 |   *   * 120

x3o3/2o5/2o5*a

. .   .   .    | 120 |   60 |   60   60 |  20  30  12
---------------+-----+------+-----------+------------
x .   .   .    |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x3o   .   .    |   3 |    3 | 2400    * |   1   1   0
x .   .   o5*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x3o3/2o   .    ♦   4 |    6 |    4    0 | 600   *   *
x3o   .   o5*a ♦  30 |   60 |   20   12 |   * 120   *
x .   o5/2o5*a ♦  12 |   30 |    0   12 |   *   * 120

x3/2o3o5/2o5*a

.   . .   .    | 120 |   60 |   60   60 |  20  30  12
---------------+-----+------+-----------+------------
x   . .   .    |   2 | 3600 |    2    2 |   1   2   1
---------------+-----+------+-----------+------------
x3/2o .   .    |   3 |    3 | 2400    * |   1   1   0
x   . .   o5*a |   5 |    5 |    * 1440 |   0   1   1
---------------+-----+------+-----------+------------
x3/2o3o   .    ♦   4 |    6 |    4    0 | 600   *   *
x3/2o .   o5*a ♦  30 |   60 |   20   12 |   * 120   *
x   . o5/2o5*a ♦  12 |   30 |    0   12 |   *   * 120

x3o3/2o5/3o5/4*a

. .   .   .      | 120 |   60 |   60   60 |  20  30  12
-----------------+-----+------+-----------+------------
x .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3o   .   .      |   3 |    3 | 2400    * |   1   1   0
x .   .   o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3o3/2o   .      ♦   4 |    6 |    4    0 | 600   *   *
x3o   .   o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x .   o5/3o5/4*a ♦  12 |   30 |    0   12 |   *   * 120

x3/2o3o5/3o5/4*a

.   . .   .      | 120 |   60 |   60   60 |  20  30  12
-----------------+-----+------+-----------+------------
x   . .   .      |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o .   .      |   3 |    3 | 2400    * |   1   1   0
x   . .   o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o3o   .      ♦   4 |    6 |    4    0 | 600   *   *
x3/2o .   o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x   . o5/3o5/4*a ♦  12 |   30 |    0   12 |   *   * 120

x3/2o3/2o5/2o5/4*a

.   .   .   .      | 120 |   60 |   60   60 |  20  30  12
-------------------+-----+------+-----------+------------
x   .   .   .      |   2 | 3600 |    2    2 |   1   2   1
-------------------+-----+------+-----------+------------
x3/2o   .   .      |   3 |    3 | 2400    * |   1   1   0
x   .   .   o5/4*a |   5 |    5 |    * 1440 |   0   1   1
-------------------+-----+------+-----------+------------
x3/2o3/2o   .      ♦   4 |    6 |    4    0 | 600   *   *
x3/2o   .   o5/4*a ♦  30 |   60 |   20   12 |   * 120   *
x   .   o5/2o5/4*a ♦  12 |   30 |    0   12 |   *   * 120

x3/2o3/2o5/3o5*a

.   .   .   .    | 120 |   60 |   60   60 |  20  30  12
-----------------+-----+------+-----------+------------
x   .   .   .    |   2 | 3600 |    2    2 |   1   2   1
-----------------+-----+------+-----------+------------
x3/2o   .   .    |   3 |    3 | 2400    * |   1   1   0
x   .   .   o5*a |   5 |    5 |    * 1440 |   0   1   1
-----------------+-----+------+-----------+------------
x3/2o3/2o   .    ♦   4 |    6 |    4    0 | 600   *   *
x3/2o   .   o5*a ♦  30 |   60 |   20   12 |   * 120   *
x   .   o5/3o5*a ♦  12 |   30 |    0   12 |   *   * 120